The quantum-mechanical energies, atomic forces and stress tensors can then be
expressed as a nonlinear function of the descriptors. In the present method, the
potential energy \(U\)  of a system of \(N_\textup {a}\)  atoms is partitioned into a
sum of local energies \(U_i\)  associated to each atom. Following the Gaussian
approximation potential (GAP) approach [1]}, each term \(U_i\)  is
written as a linear combination of kernel functions \(K\left(\mathbf {x}_i,\mathbf {x}_{b} \right)\)  weighted by coefficients \(w_{b}\) :
\( U = \sum _{i=1}^{N_\textup {a}} U_i = \sum _{i=1}^{N_\textup {a}} \sum _{b=1}^{N_b}w_{b}\, K\left(\mathbf {x}_i,\mathbf {x}_{b} \right).\) 
